Mathematical skill in individuals with Williams syndrome: evidence from a standardized mathematics battery

Williams syndrome (WS) is a developmental disorder associated with relatively spared verbal skills and severe visuospatial deficits. It has also been reported that individuals with WS are impaired at mathematics. We examined mathematical skills in persons with WS using the second edition of the Test of Early Mathematical Ability (TEMA-2), which measures a wide range of skills. We administered the TEMA-2 to 14 individuals with WS and 14 children matched individually for mental-age on the matrices subtest of the Kaufman Brief Intelligence Test. There were no differences between groups on the overall scores on the TEMA-2. However, an item-by-item analysis revealed group differences. Participants with WS performed more poorly than controls when reporting which of two numbers was closest to a target number, a task thought to utilize a mental number line subserved by the parietal lobe, consistent with previous evidence showing parietal abnormalities in people with WS. In contrast, people with WS performed better than the control group at reading numbers, suggesting that verbal math skills may be comparatively strong in WS. These findings add to evidence that components of mathematical knowledge may be differentially damaged in developmental disorders.     

Inconsistency in mathematics and the mathematics of inconsistency

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity.     

Transition and transience: Winnicott on leaving and dying

The author offers reminiscences and reflections on Donald Winnicott during the last phase of his life and considers the personal and theoretical importance for him of the topic of death. The author goes on to examine similarities and differences between Winnicott's concepts of coming into being and ceasing to be.     

Loving and leaving: Sex differences in romantic attachments

We propose a two-part generalization about sex differences in entering into and giving up romantic attachments: (1) Men tend to fall in love more readily than women; (2) women tend to fall out of love more readily than men. Evidence in support of these generalizations is derived from a longitudinal study of 231 college student dating couples. The data suggest that women are more cautious than men about entering into romantic relationships, more likely to compare these relationships to alternatives, more likely to end a relationship that seems ill fated, and better able to cope with rejection. We consider several possible explanations of these sex differences from the standpoints of psychoanalytic theory, the social and economic context of mate selection, and the socialization of men and women in the management of their own emotions. To evaluate these (and any other) explanations, further research might profitably investigate whether and to what degree these sex differences are found in other segments of the population.This research was supported by National Science Foundation grant GS-27422 to Zick Rubin. The authors are grateful to Claire Engers, Sherry Ward, and Susan Willard for their contribution to this research and to Jessie Bernard, Nancy Chodorow, George W. Goethals, Paul Rosenblatt, Ann Swidler, and Shelley Taylor for their helpful comments.     

Ideas and processes in mathematics: A course on history and philosophy of mathematics

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice mathematics teachers.     

Quitting before leaving: the mediating effects of psychological attachment and detachment on voice

This research advances understanding of the psychological mechanisms that encourage or dissuade upward, improvement-oriented voice. The authors describe how the loyalty and exit concepts from A. O. Hirschman's (1970) seminal framework reflect an employee's psychological attachment to or detachment from the organization, respectively, and they argue that psychological attachment and detachment should not be considered as separate, alternative options to voice but rather as influences on voice behavior. Findings from 499 managers in the restaurant industry show that psychological detachment (measured as intention to leave) is significantly related to voice and mediates relationships between perceptions of leadership (leader-member exchange and abusive supervision) and voice, whereas psychological attachment (measured as affective commitment) is neither a direct predictor of voice nor a mediator of leadership-voice relationships.     

Category theory and the foundations of mathematics: Philosophical excavations

The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 sections. We first show that already in the set theoretical framework, there are different dimensions to the expression foundations of. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory.Various versions of this paper have been read by many people, many of whom have made crucial comments. Needless to say, I am entirely responsible for the claims made in this paper. I would particularly like to thank, in alphabetical order, Mario Bunge, Marta Bunge, Michael Hallett, Andrew Irvine, Saunders Mac Lane, Collin McLarty, Peneloppe Maddy and Mihaly Makkai. Part of the work was done while the author was a visiting fellow at REHSEIS in Paris and at the Center for Philosophy of Science in Pittsburgh. I would like to thank everyone for his or her help and support. I gratefully acknowledge the financial support received from the SSHRC of Canada while this work was done.     

Counterfactuals and the applications of mathematics

Conclusion It has been argued that the attempt to meet indispensability arguments for realism in mathematics, by appeal to counterfactual statements, presupposes a view of mathematical modality according to which even though mathematical entities do not exist, they might have existed. But I have sought to defend this controversial view of mathematical modality from various objections derived from the fact that the existence or nonexistence of mathematical objects makes no difference to the arrangement of concrete objects. This defense of the controversial view of mathematical modality obviously falls far short of a full endorsement of the counterfactual approach, but I hope my remarks may serve to help keep such an approach a live option.     

Partition-edit-count: naive extensional reasoning in judgment of conditional probability

The authors provide evidence that people typically evaluate conditional probabilities by subjectively partitioning the sample space into n interchangeable events, editing out events that can be eliminated on the basis of conditioning information, counting remaining events, then reporting probabilities as a ratio of the number of focal to total events. Participants' responses to conditional probability problems were influenced by irrelevant information (Study 1), small variations in problem wording (Study 2), and grouping of events (Study 3), as predicted by the partition-edit-count model. Informal protocol analysis also supports the authors' interpretation. A 4th study extends this account from situations where events are treated as interchangeable (chance and ignorance) to situations where participants have information they can use to distinguish among events (uncertainty).     

Early school leaving in the lower vocational track: triangulation of qualitative and quantitative data

This study assessed the reasons for very early school leaving of boys in the lower secondary vocational track. A unique combination of quantitative and qualitative data from different sources provided background data on these boys from a national cohort study on their elementary and high school periods. In-depth interviews in which the boys reflect on their early school leaving were conducted. Four case studies are presented in which the boys' own reflections are interpreted in light of the cohort data. It was found that several factors contribute simultaneously to early school leaving; however, the emphasis lies with learning problems, lack of motivation, and problems arising from choosing the wrong vocational track. Specific personal problems also negatively affect the school career. While the boys do not feel alienated from school, they do not enjoy studying and would rather start work. This approach of combining data appears to be worth reproducing.     

Mathematical Intelligence and Mathematical Creativity: A Causal Relationship

This study investigated the causal relationship between mathematical creativity and mathematical intelligence. Four hundred thirty-nine 8th-grade students, age ranged from 11 to 14 years, were included in the sample of this study by random cluster technique on which mathematical creativity and Hindi adaptation of mathematical intelligence test were administered with 4-month time lag. Cross-lagged panel analysis was used to analyze the data. The uncorrected cross-lagged correlations appeared to show no causal relation between mathematical creativity and mathematical intelligence. But after the correction the difference in the cross-lagged correlations was found to be small and does not give guarantee of unidirectional causal relation between these two constructs. It revealed that there is a mutually reinforcing (symmetric) relationship between mathematical intelligence and mathematical creativity, i.e., mathematical intelligence causes mathematical creativity and vice-versa.     

Naive probability: a mental model theory of extensional reasoning

This article outlines a theory of naive probability. According to the theory, individuals who are unfamiliar with the probability calculus can infer the probabilities of events in an extensional way: They construct mental models of what is true in the various possibilities. Each model represents an equiprobable alternative unless individuals have beliefs to the contrary, in which case some models will have higher probabilities than others. The probability of an event depends on the proportion of models in which it occurs. The theory predicts several phenomena of reasoning about absolute probabilities, including typical biases. It correctly predicts certain cognitive illusions in inferences about relative probabilities. It accommodates reasoning based on numerical premises, and it explains how naive reasoners can infer posterior probabilities without relying on Bayes's theorem. Finally, it dispels some common misconceptions of probabilistic reasoning.